Units & measurements practice · Mixed difficulty

Errors and mistakes in measurement are primarily due to defects in the measurement process itself, such as parallax error or incorrect reading of the instrument.

Option A (If both assertion and reason are true and reason is the correct explanation of assertion) is correct because the parallax method is limited to nearby stars due to the decreasing parallactic angle with increasing distance, making accurate measurement difficult.

Option B is correct because the least count of a spherometer is calculated by dividing the pitch by the number of divisions on the circular scale, which in this case is 0.01 cm / 100 = 10^-4 cm.

Option D (Dipole moment, electrical flux, electrical field) is correct because these quantities have different dimensional formulas compared to the others.

Option D ($\left[ {\mathrm{M}{\mathrm{L}^{ - 1}}{\mathrm{T}^{ - 2}}} \right]$) is correct because modulus of rigidity is defined as shear stress divided by shear strain, and its dimensional formula is derived accordingly.

Option C ($v$) is correct because the potential energy, and therefore the maximum kinetic energy and speed, is independent of the radius R.

Option A is correct because magnetic moment is defined as current times area, so the unit is Ampere-meter squared.

Option D (A screw gauge of least count $0.001 {~mm}$) is correct because the instrument with the smallest least count provides the most accurate measurement.

Option B is correct because dimensional analysis can only derive relations with up to three unknown variables.

Option C is correct because one-watt hour is equal to 3600 joules, which is $3.6 \times 10^3$ J.

Option D (Area) is correct because for the equation to be dimensionally consistent, B must have the dimensions of wavelength squared, which is area.

Option A ($3.9 \times 10^4$) is correct because the product is rounded to the least number of significant figures in the given quantities.

Option D (Least count of instrument) is correct because precision in measurement is directly related to the least count of the measuring instrument.

Option A ($AT$) is correct because charge is the product of current and time, so its dimensions are [A][T].

Assertion is true because the permissible error in the surface area is indeed 0.6%. Reason is false because the correct formula is $\frac{\Delta A}{A}=2 \frac{\Delta r}{r}$.

Option B is correct. The term $\frac{a}{V^2}$ has the dimensions of pressure, not energy, making it dimensionally different from PV, bP, and $\frac{ab}{V^2}$ which all have dimensions of energy.

Option C is correct because the dimensions of magnetic field intensity are derived from the force on a current-carrying conductor.

Option A ($\frac{3}{2}\alpha$) is correct because the relative error in volume is 3/2 times the relative error in surface area, based on the relationship between volume and surface area of a sphere.

Option C is correct because spring constant and surface energy have the same dimensions of [ML⁰T⁻²].

Option A ($[M{L^2}{T^{ - \mathrm{2}}}]$) is correct because the dimensions of $Li^2$ represent energy, which has dimensions of mass times length squared divided by time squared.

Option B ($21.76 \times {10^{ - 19}}J$) is correct because it's the result of converting the binding energy from electron-volts to joules using the given conversion factor.

Option A ($p$) is correct because the dimensional formula of $\alpha$ matches that of pressure.

Option B ([L² A]) is correct because magnetic moment is defined as the product of current and area, and its dimensional formula is [L² A].

Option C (Personal error) is correct because parallax error arises from the observer's position and judgment while taking a reading.

Option C ($2 \times \frac{{0.01}}{{1.23}} \times 100$) is correct because the percentage error in area is twice the percentage error in the side length.

Option B ($\frac{R^2 C \lambda}{L}$) is correct because its dimensions match that of frequency (T$^{-1}$), which is verified through dimensional analysis.

Option C ([LT⁻¹]) is correct because ($\mu_0 \varepsilon_0$)^(-1/2) represents the speed of light in a vacuum, which has dimensions of length per time.

Option C ($\pm 14 \%$) is correct because it is the result of summing the weighted percentage errors of each variable in the given expression.

Option B (8%) is correct because the maximum error in kinetic energy is the sum of the percentage error in mass and twice the percentage error in speed.

Option D (${h^{0}}{c^{5}}{G^{-1}}{A^{ - 1}}$) is correct because dimensional analysis is used to find the relationship between stopping potential and Planck's constant, speed of light, gravitational constant, and ampere.